1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Rotation matrix proof

  1. Nov 3, 2015 #1
    1. The problem statement, all variables and given/known data
    Let A∈M2x2(ℝ) such that ATA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that

    A = ##
    \left( \begin{array}{cc}
    cos(\theta) & sin(\theta)\\
    sin(\theta) & -cos(\theta)\\
    \end{array} \right) ##

    It is not sufficient to show that this matrix satisfies the specified relations.
    2. Relevant equations


    3. The attempt at a solution
    Where do I start with this?! I'm supposed to get from ATA = I to the rotation matrix! I have looked at several proofs online. But they seem to either use eigenvalues and vectors (which we haven't done, so can't use them!) or don't mention the properties I've been given.

    I know a few things that might be useful.
    ##(A^T)^{-1} = (A^{-1})^T##
    And ##det(A^T) = det(A)##
    Also, I know that the matrix A is orthogonal.

    Don't know how to start!
     
  2. jcsd
  3. Nov 3, 2015 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Start with
    [tex] A = \pmatrix{a &b \\ c& d} [/tex]

    Evaluate ##P_1 = A A^T## and ##P_2 = A^T A##. You need both ##P_1 = I## and ##P_2 =I##, and those will give you several equations that the entries ##a,b,c,d## must satisfy. You also need ##\det(A) = 1##, giving you ##ad - bc = 1##.
     
  4. Nov 3, 2015 #3

    PeroK

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If you have ##a^2 + b^2 = 1##, can you show that there exists ##\theta## such that ##a = cos\theta## and ##b = sin\theta##?
     
  5. Nov 7, 2015 #4
    Got there. Thank you!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Rotation matrix proof
  1. Proofs about Matrix (Replies: 5)

  2. Matrix proof (Replies: 3)

  3. Matrix proof question (Replies: 2)

Loading...